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Recently, it is shown that there is a crucial contradiction within von Neumann’s theory [K. Nagata and T. Nakamura, Int. J. Theor. Phys. 49, 162 (2010)]. We derive a proposition concerning a quantum expected value under the assumption of the existence of the directions in a spin-1/2 system. The quantum predictions within the formalism of von Neumann’s projective measurement cannot coexist with the proposition concerning the existence of the directions. Therefore, we have to give up either the existence of the directions or the formalism of von Neumann’s projective measurement. Hence, there is a crucial contradiction within von Neumann’s theory. We discuss that this crucial contradiction makes the theoretical formulation of Deutsch’s algorithm questionable. Especially, we systematically describe our assertion based on more mathematical analysis using raw data. Our discussion, here, improves previously published argumentations very much.

Von Neumann introduces the Hilbert space and he tries to present axioms of quantum mechanics [

A quantum computer is a device for computation that makes direct use of quantum mechanical phenomena, such as superposition and entanglement, to perform operations on data. Quantum computers are different from digital computers based on transistors. Whereas digital computers require data to be encoded into binary digits (bits), quantum computation utilizes quantum properties to represent data and perform operations on these data [

Here we review to discuss that there is a crucial contradiction within von Neumann’s formalism of the quantum theory [

We review the contradiction as follows. We derive a proposition concerning a quantum expected value under the assumption of the existence of the directions in a spin-1/2 system. The quantum predictions within the formalism of von Neumann’s projective measurement (the results of measurements are

First, we review that there is a contradiction within von Neumann’s theory along with the argumentation of Ref. [

Assume a pure spin-1/2 state

We have a quantum expected value

We have

Here, the angle

We derive a necessary condition for the quantum expected value for the system in a pure spin-1/2 state lying

in the x-y plane given in (1). We derive the possible values of the product

the quantum expected value given in (1). We see that

introduce simplified notations as

where we use the orthogonality relation

sphere (the directions) with the value of

the Bloch sphere

value under the assumption of the existence of the directions (in a spin-1/2 system), that is,

when the system is in a pure state lying in the x-y plane

On the other hand, let us assume von Neumann’s projective measurement. In this case, the quantum mean value, which is the average of the results of projective measurements, is given by

We can assume as follows by Strong Law of Large Numbers,

where

where

And

By using these facts, we derive a necessary condition for the quantum mean value for the system in a pure spin-1/2 state lying in the x-y plane given in (5). The quantum mean value

Clearly, the above inequality can be saturated since, as we have said,

and

We derive the possible values of the product

proposition concerning a quantum mean value under the assumption that von Neumann’s projective mea-

surement is true (in a spin-1/2 system), that is,

of Large Numbers, we have

Thus, we have

Hence we derive the following proposition concerning von Neumann’s projective measurement

Clearly, we cannot assign the truth value “1” for two propositions (4) (concerning the existence of the directions) and (15) (concerning von Neumann’s projective measurement), simultaneously, when the system is in a pure state lying in the x-y plane. Therefore, we are in the contradiction when the system is in a pure state lying in the x-y plane.

Next, we review Deutsch’s algorithm along with Ref. [

Quantum parallelism is a fundamental feature of many quantum algorithms. It allows quantum computers to evaluate the values of a function

modulo 2. We give the transformation defined by the map

Deutsch’s algorithm combines quantum parallelism with a property of quantum mechanics known as interference. Let us use the Hadamard gate to prepare the first qubit

state

see what happens in this circuit. The input state

is sent through two Hadamard gates to give

A little thought shows that if we apply

The final Hadamard gate on the first qubit thus gives us

Realizing that

So by measuring the first qubit we may determine

In what follows, we discuss a problem of Deutsch’s algorithm. We see that the implementation of Deutsch’s algorithm is not possible if we give up either observability of a quantum state or controllability of a quantum state.

We introduce the following quantum proposition concerning controllability:

We may consider the following non-quantum-theoretical proposition:

The proposition (22) implies the validity of von Neumann’s projective measurement (observability). The proposition (22) implies

However, the validity of von Neumann’s projective measurement does not imply the proposition (22). We see that the proposition (21) is not equivalent to von Neumann’s projective measurement (observability). We see that we can assign the truth value “1” for von Neumann’s projective measurement (observability) and we can assign the truth value “0” for the proposition (21) concerning controllability.

The proposition (21) implies that

the x-y plane. The reason is as follows: Assume a pure state lying in the x-y plane as

and

proposition (21) implies that there are directions in the Hilbert space formalism of the quantum theory.

From the discussion presented in the previous, we see that the quantum proposition (21) concerning con- trollability (the directions) cannot coexist with the validity of von Neumann’s projective measurement (obser-

vability), which states

Deutsch’s algorithm shows the importance of the ability of the Hadamard gate (controllability and the existence of the directions) for quantum computation. The ability of the Hadamard gate is valid only when we assign the truth value “1” for the proposition (21) (the directions). We see that the quantum state

only when we assign the truth value “1” for the proposition (21) concerning controllability (directions) and we give up the validity of von Neumann’s projective measurement (observability). The validity of the proposition (21) implies that

We conclude that the step in which transforms the state

In conclusion, we have reviewed that there is a crucial contradiction within von Neumann’s theory. We have reexamined the quantum-theoretical formulation of Deutsch’s algorithm as the earliest quantum computer. We have resulted in the fact that the formulation has been questionable despite the fact that we have indeed had raw experimental data. We have questioned what makes observability if we accept the ability of the Hadamard gate (controllability and the directions). We have questioned what makes controllability if we accept the validity of von Neumann’s projective measurement (observability). Especially, we have systematically described our asser- tion based on more mathematical analysis using raw data. This point improves previously published argumen- tations very much.

What are new physical theories? We cannot answer it at this stage. However, we expect that our discussion in this thesis could contribute to creating new physical theories in order to explain the handing of raw experimental data, to create new information science, and to predict new unknown physical phenomena efficiently.

Koji Nagata,Tadao Nakamura, (2015) Can Von Neumann’s Theory Meet Quantum Computation?. Open Access Library Journal,02,1-6. doi: 10.4236/oalib.1101805